(, ) : R n R n R. 1. It is bilinear, meaning it s linear in each argument: that is

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1 7 Inner products Up until now, we have only examined the properties of vectors and matrices in R n. But normally, when we think of R n, we re really thinking of n-dimensional Euclidean space - that is, R n together with the dot product. Once we have the dot product, or more generally an inner product on R n, we can talk about angles, lengths, distances, etc. Definition: An inner product on R n is a function with the following properties: (, ) : R n R n R. It is bilinear, meaning it s linear in each argument: that is (c x + c x, y) = c (x, y) + c (x, y), x, x, y, c, c. and (x, c y + c y ) = c (x, y ) + c (x, y ), x, y, y, c, c.. It is symmetric: (x, y) = (y, x), x, y R n. 3. It is non-degenerate: If (x, y) = 0, y R n, then x = 0. These three properties define a general inner product. Some inner products, like the dot product, have another property: The inner product is said to be positive definite if, in addition to the above,. (x, x) > 0 whenever x 0. Remark: An inner product is also known as a scalar product (because the inner product of two vectors is a scalar). Definition: Two vectors x and y are said to be orthogonal if (x, y) = 0. Remark: The third property above, non-degeneracy, has the following meaning: the only vector x which is orthogonal to everything is the vector 0. Examples of inner products The dot product in R n is defined in the standard basis by (x, y) = x y = x y + x y + + x n y n The dot product is positive definite - all four of the properties above hold (exercise). R n with the dot product as an inner product is called n-dimensional Euclidean space, and is denoted E n.

2 Exercise: In E 3, let v = and let v = {x : x v = 0}. Show that v is a subspace of E 3. Show that dim(v ) = by finding a basis for v. In R, with coordinates t, x, y, z, we can define, (v, v ) = t t x x y y z z. This is an inner product too, since it satisfies () - (3) in the definition. But for x = (,, 0, 0) t, we have (x, x) = 0, and for x = (,, 0, 0), (x, x) = = 3, so it s not positive definite. R with this inner product is called Minkowski space. It is the spacetime of special relativity (invented by Einstein in 905, and made into a nice geometric space by Minkowski several years later). It is denoted M, and if time permits, we ll look more closely at this space later in the course. Definition: A square matrix G is said to be symmetric if G t = G. It is skew-symmetric if G t = G. Let G be an n n non-singular (det(g) 0) symmetric matrix. Define (x, y) G = x t Gy. It is not difficult to verify that this satisfies the properties in the definition. For example, if (x, y) G = x t Gy = 0 y, then x t G = 0, because if we write x t G as the row vector (a, a,..., a n ), then x t Ge = 0 a = 0, x t Ge = 0 a = 0, etc. So all the components of x t G are 0 and hence x t G = 0. Now taking transposes, we find that G t x = Gx = 0. Since G is nonsingular by definition, this means that x = 0, (otherwise the homogeneous system Gx = 0 would have non-trivial solutions and G would be singular) and the inner product is non-degenerate. You should verify that the other two properties hold as well. In fact, any inner product on R n can be written in this form for a suitable matrix G. Although we don t give the proof, it s along the same lines as the proof showing that any linear transformation can be written as x Ax for some matrix A. Examples: x y = x t Gy with G = I. For instance, if 3 x =, and y =,

3 then x y = x t Iy = x t y = (3,, ) = = 5 The Minkowski inner product has the form x t Gy with G = Diag(,,, ): t t (t, x, y, z ) x y = (t, x, y, z ) x y = t t x x y y z z z z Exercise: ** Show that under a change of basis given by the matrix E, the matrix G of the inner product becomes G e = E t GE. This is different from the way in which an ordinary matrix (which can be viewed as a linear transformation) behaves. Thus the matrix representing an inner product is a different sort of object from that representing a linear transformation. (Hint: We must have x t Gy = x t eg e y e. Since you know what x e and y e are, plug them in and solve for G e.) For instance, if G = I, so that x y = x t Iy, and ( 3 E = 3 ), then x y = x t EG E y E, with G E = ( 0 0 Exercise: ** A matrix E is said to preserve the inner product if G e = E t GE = G. This means that the recipe or formula for computing the inner product doesn t change when you pass to the new coordinate system. In E, find the set of all matrices that preserve the dot product. ). 7. Euclidean space From now on, we ll restrict our attention to Euclidean space E n. The inner product will always be the dot product. Definition: The norm of the vector x is defined by In the standard coordinates, this is equal to x = x x. ( n ) / x = x i. i= 3

4 Example: If x =, then x = ( ) + + = Proposition: x > 0 if x 0. cx = c x, c R. Proof: Exercise As you know, x is the distance from the origin 0 to the point x. Or it s the length of the vector x. (Same thing.) The next few properties all follow from the law of cosines, which we assume without proof: For a triangle with sides a, b, and c, and angles opposite these sides of A, B, and C, c = a + b ab cos(c). This reduces to Pythagoras theorem if C is a right angle, of course. In the present context, we imagine two vectors x and y with their tails located at 0. The vector going from the tip of y to the tip of x is x y. If θ is the angle between x and y, then the law of cosines reads x y = x + y x y cos θ. () On the other hand, from the definition of the norm, we have Comparing () and (), we conclude that x y = (x y) (x y) = x x x y y x + y y or x y = x + y x y x y = cos θ x y, or cos θ = x y x y () (3) Since cos θ, taking absolute values we get Theorem: x y x y () The inequality () is known as the Cauchy-Schwarz inequality. And equation (3) can be used to compute the cosine of any angle.

5 Exercises:. Find the angle θ between the two vectors v = (, 0, ) t and (,, 3) t.. When does x y = x y? What is θ when x y = 0? Using the Cauchy-Schwarz inequality, we (i.e., you) can prove the triangle inequality: Theorem: For all x, y, x + y x + y. Proof: Exercise* (Hint: Expand the dot product x + y Cauchy-Schwarz inequality, and take the square root.) = (x + y) (x + y), use the Suppose ABC is given. Let x lie along the side AB, and y along BC. Then x + y is the side AC, and the theorem above states that the distance from A to C is the distance from A to B plus the distance from B to C, a familiar result from Euclidean geometry. 5